Schoch line

In geometry, the Schoch line is a line defined from an arbelos and named by Peter Woo after Thomas Schoch, who had studied it in conjunction with the Schoch circles.

The Schoch line (cyan) passes through the point A₁.

Construction

An arbelos is a shape bounded by three mutually-tangent semicircular arcs with collinear endpoints, with the two smaller arcs nested inside the larger one; let the endpoints of these three arcs be (in order along the line containing them) A, B, and C. Let K₁ and K₂ be two more arcs, centered at A and C, respectively, with radii AB and CB, so that these two arcs are tangent at B; let K₃ be the largest of the three arcs of the arbelos. A circle, with the center A₁, is then created tangent to the arcs K₁,K₂, and K₃. This circle is congruent with Archimedes' twin circles, making it an Archimedean circle; it is one of the Schoch circles. The Schoch line is perpendicular to the line AC and passes through the point A₁. It is also the location of the centers of infinitely many Archimedean circles, e.g. the Woo circles.[1]

Radius and center of A₁

If r = AB/AC, and AC = 1, then the radius of A₁ is

${\displaystyle \rho ={\frac {1}{2}}r\left(1-r\right)}$

and the center is

${\displaystyle \left({\frac {1}{2}}r\left(-1+3r-2r^{2}\right)~,~r\left(1-r\right){\sqrt {\left(1+r\right)\left(2-r\right)}}\right).}$[1]

References

1. Dodge, Clayton W.; Schoch, Thomas; Woo, Peter Y.; Yiu, Paul (1999), "Those ubiquitous Archimedean circles" (PDF), Mathematics Magazine, 72 (3): 202–213, doi:10.2307/2690883, JSTOR 2690883, MR 1706441.

Further reading

• Okumura, Hiroshi; Watanabe, Masayuki (2004), "The Archimedean circles of Schoch and Woo" (PDF), Forum Geometricorum, 4: 27–34, MR 2057752.

External links

• van Lamoen, Floor. "Schoch Line." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein". Retrieved 2008-04-11.